Combinatorial Game Theory
who wins, and by how much
Concept

Reduced canonical form — where it appears

The canonical form with one clause added: an option is also deleted when it differs from another by an infinitesimal. It is what a position is worth once the small change is thrown away, and it adds without composing — the sum of two of them sometimes needs reducing again.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Add, then reduce again. The arithmetic the homomorphism promises, measured: summing two reduced forms gives a reduced form on 88 per cent of pairs and needs a second reduction on the rest.

Add, then reduce again

The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.

sums · Reduced form Eight ways to name it, and none of them works. Candidate rules for which option the second reduction deletes, scored on every pair where it deletes exactly one. The best reaches four in five and none is exact.

The option nothing names

The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.

sums · Reduced form Not a domination, in the order the rung below meant. The second pass's deletions scored as dominations in two orders: the partial order on games, and the order on stops.

Not a domination, in that order

The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

sums · Reduced form The test, scored. The recognition test run on every deletion the second reduction makes, against what actually happens.

A side about to lose its move

A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.

sums · Reduced form A cross in the table. Pairs needing a second reduction, by the colder temperature and the gap between the two.

A cross in the table

Which pairs need a second reduction had never been asked. Sorted by the two temperatures the answer is a cross — the whole gap-of-a-quarter column and the whole colder-is-three-quarters row — and it is exactly necessary on all 431 with no exception, and wrong 477 times in the other direction.

sums · Reduced form

Named alongside it

The objects these essays reach for when they reach for this one.

Canonical formDisjunctive sumEnumerationInfinitesimalStopsDominanceTemperatureApproximationDay threeDominated optionInvariantPartial order

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